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(1) Show that the following space of infinite sequences c

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(1)

CLASSICAL APPOXIMATION HOMEWORKS

Due date is November 18, 2019

(1) Show that the following space of infinite sequences c

0

:= {(x

1

, x

2

, . . .) : x

i

∈ R, lim

i→∞

x

i

= 0}

with the norm max

i­1

|x

i

| is not strictly convex.

(2) Find the best approximation in L

2

([0, 1]) for the function f (x) = x with respect to the space spanned by v

1

(x) = e

x

i v

2

(x) = e

2x

.

(3) Suppose that none of the points a

1

, a

2

, . . . , a

n

is in the interval [a, b]. Show that then

span



1 x − a

1

, 1

x − a

2

, . . . , 1 x − a

n



is a Haar space in C([a, b]).

(4) For what values of a < b the space spanned by the functions (a) {1, cos(x), cos(2x), . . . , cos(nx)}

(b) {sin(x), sin(2x), . . . , sin(nx)}

is a Haar space in C([a, b])?

(5) Let D be the unit sphere,

D = {~ x ∈ R

s

: k~ xk

2

= 1 } ⊂ R

s

.

For what values of s and n one can find Haar spaces of dimension n that are subspaces of C(D)?

(6) Find a polynomial p of degree ¬ 3 that minimizes sup

−1¬x¬1

| |x| − p(x) |.

(7) Find a trigonometric polynomial of the form v(t) = a

0

+ a

1

sin t + b

1

cos t

that best approximates the function sin(t/2) in the uniform norm on the interval [−π, π].

(8) In the set of polynomials p of degree ¬ n such that p(0) = 1 find p

that has minimal uniform norm on [1, 2].

(9) Let p be a polynomial of degee at most n such that kpk

C([−1,1])

¬ 1. Show that then for any |x| ­ 1 we have |p(x)| ¬ |T

n

(x)|.

(10) Show that among all the polynomials of degree at most n such that p

0

(1) = A, the polynomial AT

n

/n

2

has the minimal uniform norm in [−1, 1].

Due date is December 9, 2019

(11) Let the operator L : C([a, b]) → C([a, b]) be given as (Lf )(x) =

n

X

i=1

f (x

i

)g

i

(x),

1

(2)

2

where a ¬ x

1

< · · · < x

n

¬ b and g

i

∈ C([a, b]). Show that L is positive if and only if all the functions g

i

assume nonnegative values only.

(12) Let L : C([a, b]) → C([a, b]) be a positive linear operator satisfying Lw = w for all polynomials of degree at most 2. Show that then Lf = f for all f ∈ C([a, b]).

(13) Show that if f is a polynomial of degree at most k then the same property possess all the Bernstein polynomials B

n

f.

(14) Let E be the family of projections L : C([a, b]) → P

n+1

, and ˆ E be the family of projections ˆ L : C([−1, 1]) → P

n+1

. Show that

L

inf

n∈E

kL

n

k = inf

Lˆn∈ ˆE

k ˆ L

n

k.

(15) Let L : C([−1, 1]) → P

3

be the interpolation operator corresponding to some points x

i

, i = 0, 1, 2. Show that

−1¬x0

min

<x1<x2¬1

kLk = 5 4 ,

and for the Chebyshev points we have kLk = 5/3. What is kL

2

k for the equ- ispaced points −1, 0, 1?

(16) Show the following properties of the Chebyshev polynomial T

n

. (a) (1 − x

2

)T

n00

(x) − xT

n0

(x) + n

2

T

n

(x) = 0

(b) T

2n

(x) = T

n

(2x

2

− 1) (c) T

n

(T

m

) = T

nm

(17) Let f (x) = x

3

. Find possibly minimal n such that B

n

f approximates f with error at most 10

−8

with respect to the uniform norm on [0, 1]?

(18) Show that if f i f

0

are continuous on [0, 1] then for any  > 0 there is a polynomial p such that kf − pk ¬  and kf

0

− p

0

k ¬ , where the norm is uniform on [0, 1].

(19) Let X = L

2

([0, 1]), f (x) = x

m

and v

k

(x) = x

pk

, k = 1, 2, . . . , n, where 0 ¬ m < p

1

< p

2

< · · · < p

n

.

Let V

n

= span(v

1

, v

2

, . . . , v

n

). Show that dist(f, V

n

)

2

= 1

2m + 1

n

Y

k=1

m − p

k

m + p

k

+ 1

!2

.

Hint.

det



1 a

k

+ b

k

n k,l=1

!

=

Q

k>l

(a

k

− a

l

)(b

k

− b

l

)

Q

k,l

(a

k

+ b

l

) . (20) Let 0 ¬ m < p

1

< p

2

< · · · . Show that

n→∞

lim

n

Y

k=1

m − p

k

m + p

k

+ 1

!2

= 0 if and only if

X

k=2

1 p

k

= ∞.

(3)

3

(21) (M¨ unz theorem I) Show that the space spanned by the functions v

k

(x) = x

pk

, where 0 ¬ p

1

< p

2

< · · · , is dense in L

2

([0, 1]) if and only if

Pk=2

1/p

k

= ∞.

Hint. Use Problems 19 i 20 and the fact that algebraic polynomials are dense in L

2

([0, 1]).

(22) (M¨ unz theorem II) Show that the space spanned by the functions v

k

(x) = x

pk

, where 0 ¬ p

1

< p

2

< · · · , is dense in C([0, 1]) if and only if p

1

= 0 and

P

k=2

1/p

k

= ∞. Hint. Use Problem (21).

(23) Is the space spanned by 1, x

p1

, x

p2

, . . . , x

pn

, . . ., where p

n

are successive primes,

dense in C([0, 1])?

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